268 XIV INTEGRATION IN LOCALLY COMPACT GROUPS Let /be any function belonging to tf(G), and put h(jf) = f /(*{) da'«) for n(x) = x". By virtue of (14.4.2) we have [f(u-\x))d*(x) = (d^(±)(f(u-\x)u-\^d^(^ and since u"1^) = u"~l(£) by definition, f /(iT'OOiT1®) <fa'«) = (mod(iO) ' f /(""'WO <**'(«) ^(w-'Oc)). Since J h(ulf'1(xir)) da"(x") = (mod(w")) J *(*") da,"(x"\ we obtain the required result. (1 4.4.7) For each xeG we have where i* denotes the automorphism sf\-*x~ls'x of G'. Apply (14.4.6) with u = ix9 and use (14.3.7). In particular, if x' e G', we have AG(jcO = AG,(^). PROBLEMS 1. Let G be a locally compact group. A measure jn on G is said to be relatively left- (resp. right-) invariant if for each s e G there exists a complex number x($) ^ 0 such that y(s)[ji = xWp fresP- S(^)/x = xC5)^)- Show that x is a continuous homomorphism of G into the multiplicative group C*, and that ^ is of the form a*£~l • ft, where fi is a left (resp. right) Haar measure on G and a is a complex number =^0 (Section 14.3, Problem 4(a)). Consider the converse of this result, and deduce that every relatively left-invariant measure on G is also relatively right-invariant. 2. Let G be a locally compact group, H a closed subgroup of G, a a left Haar measure on G, £ a left Haar measure on H, TT the canonical mapping of G onto G/H. (a) For each function / e Jf (G), show that there exists a unique function / fr e Jf (G/H) such that /b(7r(x)) = f /(*£) </£(£). Conversely, for each A e ^f (G/H), there exists a JH function/e «#XG) such that/I7 = /z. (If C is the support of A, consider a compact subset K of G such that 7r(K) — C (12.10.9) and a function # e ^T(G) such that #(#) > 0 for all x e K, and observe that (g • (h ° ?r))b — g* - h. (b) A measure /x ^ 0 on G/H is said to be relatively invariant under G if for eachG). Then modG(w) =